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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.PE.88
Chapter 4, Problem 4.PE.88

Finding Indefinite Integrals
Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ cos³ 𝓍/2 d𝓍

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1
Rewrite the integral to make it easier to handle. Use the identity for powers of cosine: express \(\cos^3\left(\frac{x}{2}\right)\) as \(\cos\left(\frac{x}{2}\right) \cdot \cos^2\left(\frac{x}{2}\right)\).
Recall the Pythagorean identity: \(\cos^2\theta = 1 - \sin^2\theta\). Substitute this into the integral to get \(\int \cos\left(\frac{x}{2}\right) \left(1 - \sin^2\left(\frac{x}{2}\right)\right) dx\).
Use substitution to simplify the integral. Let \(u = \sin\left(\frac{x}{2}\right)\), then find \(du\) in terms of \(dx\). Since \(u = \sin\left(\frac{x}{2}\right)\), differentiate to get \(du = \frac{1}{2} \cos\left(\frac{x}{2}\right) dx\), or equivalently, \(\cos\left(\frac{x}{2}\right) dx = 2 du\).
Rewrite the integral in terms of \(u\): it becomes \(\int (1 - u^2) \cdot 2 du = 2 \int (1 - u^2) du\). This integral is straightforward to evaluate.
Integrate term-by-term: \(2 \int (1 - u^2) du = 2 \left( u - \frac{u^3}{3} \right) + C\). Finally, substitute back \(u = \sin\left(\frac{x}{2}\right)\) to express the antiderivative in terms of \(x\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Indefinite Integral

An indefinite integral represents the most general antiderivative of a function, expressed with a constant of integration (C). It reverses differentiation and provides a family of functions whose derivative is the integrand.
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Integration Techniques for Powers of Trigonometric Functions

Integrating powers of trigonometric functions often requires rewriting the integrand using identities or substitution. For example, expressing cos³(x/2) as cos(x/2)·cos²(x/2) and then using the identity cos²θ = 1 - sin²θ simplifies the integral.
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Verification by Differentiation

After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the indefinite integral and helps identify any errors in the integration process.
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Related Practice
Textbook Question

Initial Value Problems

Solve the initial value problems in Exercises 89–92.


dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec² s/10 ds

Textbook Question

Applications


Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

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∫ ( 3√ t + 4/t² ) dt